Let m,n>1 be independent integers, and let X,Y \subset [0,1] be two closed sets that are times m and times n invariant, respectively. In 1970 Furstenberg conjectured that for every invertible affine map g, the Hausdorff dimension of g(X)\cap Y is at most the maximum of dim X + dim Y - 1 and 0. In 2016 Shmerkin and Wu (independently) proved the Conjecture to be correct. We shall present a generalization of this result: For every carpet in a class of self affine carpets related to the pair (m,n), the dimension of the intersection of the carpet with any affine line with non-zero and finite slope can be bounded by a corresponding bound. Our proof relies on an adaptation of Wu's ergodic proof of Furstenberg's Conjecture.